Q4) Let X={a, b, c, d} with the topology T={Ø, X, {a}, {b}, {a,b}, {b,c,d}}. Let f: (X,T)→(X,T) such that f(a)=f(c)=Db, f(b)=Dd and f(d)=c. Then a. f is continuous at d b. f is continuous on X c. f is continuous at c d. f is discontinuous every where

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Q: (a) Prove that f is a one-to-one correspondence. (b) Prove that ray AF is between rays AB and AE if…

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Q: Let X:= (a, b.c), n=(1,2,3.4), Ty:= (0,Y, {1,3), [3,4}, {3), [1,3,4}}. Let fx-Y. f:= {(a,3). (b, 3).…

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. * None…

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Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

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Q: Let f = 12r + kry + Jryz re-express f O f(r,y,z) = (x, ry, Tyz+) O {r,y) = (x, ry, ryz*) O f(x,y) =…

A: GIVEN f= ix+kxy+jxyz4     ....(1)

Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

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Q: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Let f: R" x R → R define as follows f(x,y) = a, then f is

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Q: AP1) Prove that f: E → R; x x² is uniformly continuous if E is bounded. AP2) Prove that f : R R; x x…

A: Definition: A function fx said to be uniformly continuous, if for every ε>0, there exists δ>0…

Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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Q: .Consider the function f(7,4) = { xy x²+4y? if (x, y) # (0,0) if (x, y) = (0,0) O Is f continuous at…

A: Since the given function is of type function of several variable therefore

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Q: Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given…

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Q: Let (f (u, v,w) and (u = u(x, y, z), v = v(x, y, z), w = w(x, y, z) be differentiable functions)…

A: Given function f(u, v, w) and u = u(x, y ,z),  v = v(x, y, z) , w = w(x,y,z) are differentiable…

Q: Let X = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

Q: (a) Let Ta be the particular point topology on X = {a,b,c, d}, and f:(X,ta) → (X, Ta) is defined by…

A: In the given question given that ζa be a particular point topology on X={a,b,c,d}.so first of all we…

Q: Suppose that R³ is given the norm ||(x, y, z) || = |x|+|y| + |2| and let S = {(x, y, z) = R³: 2x + y…

A: Given that R3 has a norm ||(x,y,z)||=|x|+|y|+|z|.The given set is S={(x,y,z)∈R3:2x+y−3z=0}.Now…

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Q: Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A: X = { a, b, c } , Y = { 1, 2, 3 } T = { Y, ∅,  {1, 2} } f = { (a, 1), (b, 3), (c, 3) }

Q: 3: Let a function d: R* x R* → R defined by d(x, y) = E-\xi - yıl Vx = (x1,...,x4), y = (y1,...,y4)…

A: Here we will prove that d is a metric function.

Q: Let X = {0, r, t, w}, Y = {e,g, i, k, m}, and Z = {2,3,4,5, 7,9}. Let f : X → Y where f = {(0, g),…

A: Given X = {o, r, t, w} Y = {e, g, i, k, m} Z = {2, 3, 4, 5, 7, 9}

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Q: Prove that if f : V → W is differentiable at a point a ∈ V it must be continuous there.

A: I have proved limiting value is equal to functional value

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Q: Let (X, A, µ) and (Y, B, v) be measure spaces. Prove that (a) if E E A x B then y-section of E is…

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Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

A: Given that T be the co-finite topology on ℝ and d0x,y=x-y To prove that the function f:ℝ,T→ℝ,d0…

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A: According to our guidelines I can answer only Question no. 4(a).

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Q: Let C[a, b] denotc the set of continuous functions f : [a, b] → R. Given any two functions S,9 €…

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Q: 5) Let 7 be the euclidean topology on R and 7' be the finite-closed topology on R. Define f : (R,…

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Q: Define f: Z x Z→ Z x Z by f((m, n)) = (3, 2n). Let A = {(n, 0): n ∈Z}. Fina f(A).

A: EXPLANATION : Given function  f:ℤ×ℤ→ℤ×ℤ defined by fm,n=(3,2n). given set A={(n,0):n∈ℤ}. putting the…

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